Theorem ssrabdv 3280

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	|- ( ph -> B C_ A )
ssrabdv.2	|- ( ( ph /\ x e. B ) -> ps )

Assertion

Ref	Expression
ssrabdv	|- ( ph -> B C_ { x e. A | ps } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 |- ( ph -> B C_ A )
2		ssrabdv.2	. . 3 |- ( ( ph /\ x e. B ) -> ps )
3	2	ralrimiva 2581	. 2 |- ( ph -> A. x e. B ps )
4		ssrab 3279	. 2 |- ( B C_ { x e. A | ps } <-> ( B C_ A /\ A. x e. B ps ) )
5	1, 3, 4	sylanbrc 417	1 |- ( ph -> B C_ { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   A.wral 2486   {crab 2490   C_ wss 3174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-in 3180  df-ss 3187

This theorem is referenced by:  perfectlem2  15587